WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

Transmission
1.00
0.95
0.90
0.85
Structure 1
0.80
0.75
1480 1500 1520 1540 1560
Structure 2
Structure 3
Structure 4
1580 1600 1620
Wavelength (nm)
Figure 7: Total energy transmitted through top and bottom output ports relative to the input wave
energy for the 4 structures in Figures 5 and 6.
the first two structures is basically kept but enlarged aft of the structure. As can been seen this allows
for placing an extra reflector aft of the waveguide at a distance commensurable with the wavelength.
By comparing the response for structure 4 and structure 2 in Figure 7, it is noticed that this difference
of the otherwise quite similar structures has caused a considerable improvement of the 2D transmission.
5.4. T-splitter - TE polarization
The second example is for TE-polarized waves. The basic system dimensions and material properties
are taken from [3] in which propagation losses in straight waveguides with simple bends were studied
experimentally. In this case w = 445nm and n = 3.5. The optimized structures in Figure 8 are both
obtained by simulating dissipation in the air.
Figure 8 (top) (structure 1) shows the optimized structure obtained by maximizing the transmission
for a wavelength of 1.55µm. For TE waves this corresponds to a significantly shorter wavelength compared
to the width of the waveguide than for the TM wave example. This difference is noticeable in
the optimized structures which possess structural details at a corresponding smaller scale. It is noted
that the transmission is almost 100% near the target frequency. The design shown in Figure 8 (bottom)
(structure 2) was obtained by attempting to get a high transmission for a broader frequency range. This
was done by modifying the objective in the optimization problem (Eq. (4)) to include the transmission
at two distinct frequencies:
max min Φ1(ω1), Φ1(ω2), Φ2(ω1), Φ2(ω2). (8)
The two target frequencies ω1 and ω2 are not kept fixed during the optimization procedure, but are
repeatedly updated (e.g. every 10-20 iterations) so that they correspond to the critical frequencies with
minimum transmission. The critical frequencies are identifying by fast-frequency-sweeps using Padé
approximants [6]. Naturally, the number of frequencies in Eq. (8) can be increased which is necessary if
a larger frequency range is considered.
Figure 9 shows a comparison of the performance of the two designs. As noted the bandwidth of
structure 2 is slightly better than that of structure 1, but this has been obtained at the expense of the
high transmission near 1.55µm. Including more frequencies in the optimization does not change this
significantly.
6. Conclusions
We have demonstrated the use of topology optimization to design low-loss photonic circuit components
such as bends and splitters.
Previously it has been shown that such components in photonic crystal waveguides can effectively
be designed with topology optimization, leading to low losses in large frequency ranges. In this paper
we have demonstrated design of similar low loss components also in photonic wire waveguides. These
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